Pound chicken breasts to 1/2-inch thickness. Season both sides with salt and pepper. Dredge lightly in flour, shaking off excess.
Heat olive oil and butter in a large skillet over medium heat. Sear chicken 3 to 4 minutes per side until golden. Transfer to a plate.
Add white wine to the skillet, scraping up browned bits. Simmer 2 minutes. Add chicken stock and simmer 3 to 4 minutes to reduce slightly.
In a separate saucepan, heat 2 tablespoons olive oil over medium heat. Saute onion, garlic, and carrot if using until softened, about 5 to 7 minutes. Stir in crushed tomatoes, oregano, salt, pepper, and a pinch of sugar if using. Simmer uncovered 10 to 15 minutes until thickened.
Return chicken to the skillet with the wine and stock. Spoon tomato sauce generously over each breast. Top with buffalo mozzarella slices.
Cover and simmer on low 3 to 5 minutes until cheese is melted, or broil 2 to 3 minutes until bubbling and lightly browned.
Cook angel hair in well-salted boiling water until al dente. Reserve 1/2 cup pasta water. In the same pot, melt butter over medium heat, add garlic, and saute until fragrant about 1 minute. Toss pasta in garlic butter, adding reserved pasta water a little at a time until well coated. Season to taste.
Plate pasta and top with chicken. Spoon pan sauce over everything and garnish with fresh parsley.